These are Homer's Powers in The Simpsons Game.
Homer's burp is his super-attack. He needs to collect power balls to get enough power to do a massive burp to defeat near enemies. On the PlayStation 3 and/or Xbox 360 versions, Homer's power burp is fire while on the PS2 version of the game, his burp is just extended green gas.
"Homer Ball" is one of Homer's powers in The Simpsons Game. He needs to eat food to gain enough power to become a big ball that can roll on the floor. As Homer Ball, Homer can use the "Homer Ball boost" to boost into enemies and walls, and also the "Homer Ball super-boost", in which he can jump and splat on the floor. While doing this, he loses power, so he has to eat to stay in his Homer Ball form. The first level in which Homer uses it is "The Land of Chocolate", but he discovers it in "Around the World in 80 Bites".
Homer has another power called "Heli-Homer". He can inhale helium to become a gas ball and float onto the air while he has enough power. As Heli-Homer, Homer can use the "Heli-Homer boost" to boost into the air and cross certain areas quicker. The first level in which Homer uses it is "Shadow of the Colossal Donut", one of the levels that introduces the family's power upgrades.
Homer has one more power called the "Gumi-Homer". He needs to eat Gummi Venus de Milo to gain enough power to become a big, green gumi ball that can shoot gumi-balls to further enemies and use the "Gumi-Boost" to do a boost on the floor to launch gumi pieces to the enemies. While doing this, Homer loses power, so he has to eat Gumis to keep transformed into Gumi-Homer. The first level in which Homer uses it is "NeverQuest", but he learns how to use it before in the game engine, where Professor Frink teaches him about how to properly use the power. In this form, Homer is always facing forward; the player can't see his face while he's in this form.